Beurling Spectra of Functions on Locally Compact Abelian Groups

TL;DR

This paper investigates the spectral properties of functions on locally compact abelian groups, characterizing spectra for polynomials, primary ideals in Beurling algebras, and conditions for boundedness of functions and their indefinite integrals.

Contribution

It introduces new spectral notions for functions on abelian groups, characterizes primary ideals in Beurling algebras, and provides criteria for boundedness based on spectral conditions.

Findings

Spectra of polynomials on groups are determined.

Primary ideals of certain Beurling algebras are characterized.

Spectral conditions ensure boundedness of functions and their indefinite integrals.

Abstract

Let $G$ be a locally compact abelian topological group. For locally bounded measurable functions $\varphi: G\to\Bbb {C}$ we discuss notions of spectra for $\varphi$ relative to subalgebras of $L^{1}(G)$. In particular we study polynomials on $G$ and determine their spectra. We also characterize the primary ideals of certain Beurling algebras $L_{w}^{1}(\Bbb Z)$ on the group of integers $\Bbb Z$. This allows us to classify those elements of $L_{w}^{1}(G)$ that have finite spectrum. If $\varphi$ is a uniformly continuous function whose differences are bounded, there is a Beurling algebra naturally associated with $\varphi$. We give a condition on the spectrum of $\varphi$ relative to this algebra which ensures that $\varphi$ is bounded. Finally we give spectral conditions on a bounded function on $\Bbb R$ that ensure that its indefinite integral is bounded.